Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra PDF
Preview Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra
Undergraduate Texts in Mathematics David A. Cox John Little Donal O'Shea Ideals, Varieties, and Algorithms An Introduction to Computational Algebraic Geometry and Commutative Algebra Fourth Edition Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics SeriesEditors: SheldonAxler SanFranciscoStateUniversity,SanFrancisco,CA,USA KennethRibet UniversityofCalifornia,Berkeley,CA,USA AdvisoryBoard: ColinAdams,WilliamsCollege DavidA.Cox,AmherstCollege PamelaGorkin,BucknellUniversity RogerE.Howe,YaleUniversity MichaelOrrison,HarveyMuddCollege JillPipher,BrownUniversity FadilSantosa,UniversityofMinnesota Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an apprecia- tionofinterrelationsamongdifferentaspectsofthesubject.Theyfeatureexamples thatillustratekeyconceptsaswellasexercisesthatstrengthenunderstanding. Moreinformationaboutthisseriesathttp://www.springer.com/series/666 David A. Cox • John Little • Donal O’Shea Ideals, Varieties, and Algorithms An Introduction to Computational Algebraic Geometry and Commutative Algebra Fourth Edition 123 DavidA.Cox JohnLittle DepartmentofMathematics DepartmentofMathematics AmherstCollege andComputerScience Amherst,MA,USA CollegeoftheHolyCross Worcester,MA,USA DonalO’Shea President’sOffice NewCollegeofFlorida Sarasota,FL,USA ISSN0172-6056 ISSN2197-5604 (electronic) UndergraduateTextsinMathematics ISBN978-3-319-16720-6 ISBN978-3-319-16721-3 (eBook) DOI10.1007/978-3-319-16721-3 LibraryofCongressControlNumber:2015934444 MathematicsSubjectClassification(2010):14-01,13-01,13Pxx SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland1998,2005,2007,2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternational PublishingAGSwitzerlandispartofSpringerScience+Business Media(www. springer.com) To Elaine, for herloveand support. D.A.C. To thememoryofmyparents. J.B.L. To Maryandmychildren. D.O’S. Preface We wrote this book to introduce undergraduates to some interesting ideas in algebraicgeometryandcommutativealgebra.Foralongtime,thesetopicsinvolved alotofabstractmathematicsandwereonlytaughtatthegraduatelevel.Theircom- putationalaspects, dormantsince the nineteenth century,re-emergedin the 1960s with Buchberger’s work on algorithms for manipulating systems of polynomial equations.The developmentofcomputersfastenoughto runthese algorithmshas madeitpossibletoinvestigatecomplicatedexamplesthatwouldbeimpossibletodo byhand,andhaschangedthepracticeofmuchresearchinalgebraicgeometryand commutativealgebra.Thishasalsoenhancedtheimportanceofthesubjectforcom- puterscientists andengineers,whonow regularlyuse these techniquesin a whole rangeofproblems. It is our belief that the growing importance of these computational techniques warrantstheirintroductionintotheundergraduate(andgraduate)mathematicscur- riculum.Manyundergraduatesenjoytheconcrete,almostnineteenthcentury,flavor thata computationalemphasisbringstothesubject.Atthesame time,onecando some substantial mathematics, including the Hilbert Basis Theorem, Elimination Theory,andtheNullstellensatz. Prerequisites Themathematicalprerequisitesofthebookaremodest:studentsshouldhavehada courseinlinearalgebraandacoursewheretheylearnedhowtodoproofs.Examples ofthelattersortofcourseincludediscretemathandabstractalgebra.Itisimportant to note that abstract algebra is not a prerequisite. On the other hand, if all of the students have had abstract algebra, then certain parts of the course will go much morequickly. Thebookassumesthatthestudentswillhaveaccesstoacomputeralgebrasys- tem.AppendixCdescribesthefeaturesofMapleTM,Mathematica(cid:2),Sage,andother computeralgebrasystemsthataremostrelevanttothetext.Wedonotassumeany prior experience with computer science. However, many of the algorithms in the vii viii Preface book are described in pseudocode, which may be unfamiliar to students with no backgroundinprogramming.AppendixBcontainsacarefuldescriptionofthepseu- docodethatweuseinthetext. HowtoUsetheBook Inwritingthebook,wetriedtostructurethematerialsothatthebookcouldbeused in a variety of courses, and at a variety of differentlevels. For instance, the book couldserveasabasisofasecondcourseinundergraduateabstractalgebra,butwe think that it just as easily could provide a credible alternative to the first course. Although the book is aimed primarily at undergraduates,it could also be used in variousgraduatecourses,withsomesupplements.Inparticular,beginninggraduate coursesinalgebraicgeometryorcomputationalalgebramayfindthetextuseful.We hope,ofcourse,thatmathematiciansandcolleaguesinotherdisciplineswillenjoy readingthebookasmuchasweenjoyedwritingit. Thefirstfourchaptersformthecoreofthebook.Itshouldbepossibletocover them in a 14-week semester, and there may be some time left over at the end to exploreotherpartsofthetext.Thefollowingchartexplainsthelogicaldependence ofthechapters: 1 2 §9 3 §5 §6 §1 4 6 §7 8 5 7 §7 9 10 Preface ix Thetableofcontentsdescribeswhatiscoveredineachchapter.Asthechartin- dicates,thereareavarietyofwaystoproceedaftercoveringthefirstfourchapters. Thethreesolidarcsandonedashedarcinthechartcorrespondtospecialdependen- cies that will be explained below. Also, a two-semester course could be designed thatcoverstheentirebook.Forinstructorsinterestedinhavingtheirstudentsdoan independentproject,wehaveincludedalistofpossibletopicsinAppendixD. FeaturesoftheNewEdition Thisfourthedition incorporatesseveralsubstantialchanges.In some cases, topics have been reorganized and/or augmented using results of recent work. Here is a summaryofthemajorchangestotheoriginalninechaptersofthebook: • Chapter2:Wenowdefinestandardrepresentations(implicitinearliereditions) and lcm representations(new to thisedition).Theorem6 from§9 playsan im- portantroleinthebook,asindicatedbythesolidarcsinthedependencecharton thepreviouspage. • Chapter3:WenowgivetwoproofsoftheExtensionTheorem(Theorem3in§1). Theresultantprooffromearliereditionsnowappearsin§6,andanewGröbner basisproofinspiredbySCHAUENBURG (2007)ispresentedin§5.Thismakesit possibleforinstructorstoomitresultantsentirelyiftheychoose.However,resul- tantsareusedintheproofofBezout’sTheoreminChapter8,§7,asindicatedby thedashedarcinthedependencechart. • Chapter4:Thereareseveralimportantchanges: – In§1wepresentaGröbnerbasisproofoftheWeakNullstellensatzusingideas fromGLEBSKY(2012). – In§4wenowcoversaturationsI:J∞inadditiontoidealquotientsI:J. – In §7 we use Gröbner bases to prove the Closure Theorem (Theorem 3 in Chapter3,§2)followingSCHAUENBURG (2007). • Chapter5:Wehaveaddedanew§6onNoethernormalizationandrelativefinite- ness.Unliketheprevioustopics,theproofsinvolvedinthiscasearequiteclassi- cal.Buthavingthismaterialtodrawonprovidesanotherilluminatingviewpoint inthestudyofthedimensionofavarietyinChapter9. • Chapter6:ThediscussionofthebehaviorofGröbnerbasesunderspecialization in §3 has been supplemented by a brief presentation of the recently developed conceptofaGröbnercoverfromMONTESandWIBMER(2010).Wewouldlike tothankAntonioMontesfortheGröbnercovercalculationreportedin§3. Inthebiggestsinglechange,wehaveaddedanewChapter10presentingsome oftheprogressofthe past25yearsin methodsforcomputingGröbnerbases(i.e., sincetheimprovedBuchbergeralgorithmdiscussedinChapter2,§10).Wepresent Traverso’sHilbertdrivenBuchbergeralgorithmforhomogeneousideals,Faugère’s F algorithm,and a brief introductionto the signature-basedfamily of algorithms 4 includingFaugère’sF .Thesenewalgorithmicapproachesmakeuseofseveralin- 5 terestingideasfrompreviouschaptersandleadthereadertowardsomeofthenext steps in commutative algebra (modules, syzygies, etc.). We chose to include this
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